Question

Let $p\left(x\right)$ be the fifth degree polynomial such that $p\left(x\right)+1$ is divisible by $(x-1)$ and $p\left(x\right)-1$ is divisible by $(x+1)$.Then find the value of ${\int }_{-10}^{10}p\left(x\right)dx$

Answer 1

$p\left(x\right)+1$ is visible by $x-1$

$\Rightarrow p\left(1\right)+1=0\Rightarrow p\left(1\right)=-1$

$p\left(x\right)-1$ is divisible by $(x+1)$

$\Rightarrow p(-1)-1=0\Rightarrow p(-1)=1$

Hence $p(-x)=-p\left(x\right)$

Hence, $p\left(x\right)$ is odd function.

$\therefore$ ${\int }_{-10}^{10}p\left(x\right)dx=0$